modeling and motion control of mobile robot for lattice type welding

Modeling and Motion Control of Mobile Robotfor Lattice Type Welding Yang Bae Jeon*, Sang Bong Kim Department of Mechanical Engineering, College, Pukyong National University, Korea Soon S

Modeling and Motion Control of Mobile Robot

for Lattice Type Welding Yang Bae Jeon*, Sang Bong Kim

Department of Mechanical Engineering, College, Pukyong National University, Korea

Soon Sil Park

Renault Samsung Motors Co., Ltd 185, Shinho-dong, Kangseo-gu, Pusan 618-722, Korea

This paper presents a motion control method and its simulation results of a mobile robot for

a lattice type welding Its dynamic equation and motion control methods for welding speed and seam tracking are described The motion control is realized in the view of keeping constant welding speed and precise target line even though the robot is driven for following straight line

or curve The mobile robot is modeled based on Lagrange equation under nonholonomic constraints and the model is represented in state space form The motion control of the mobile robot is separated into three driving motions of straight locomotion, turning locomotion and torch slider control For the torch slider control, the proportional-integral-derivative (PID) control method is used For the straight locomotion, a concept of decoupling method between input and output is adopted and for the turning locomotion, the turning speed is controlled according to the angular velocity value at each point of the corner with range of 90° constrained

to the welding speed The proposed control methods are proved through simulation results and these results have proved that the mobile robot has enough ability to apply the lattice type welding line

Key Words:Mobile Robot, Motion Control, Nonholonomic Constraints, Decoupling Method

Nomenclature

-b : Distance between driving wheel and

symmetry axis

d :Distance from Po to mass center of

mobile robot

D :Viscous friction

Ie :Inertia moment of mobile robot

excluding driving wheels and rotors of

motors on a vertical axis through

inter-section between symmetry axis and

driving wheel axis

1m : Inertia moment of wheel and motor

rotor on wheel diameter

I w : Inertia moment of wheel and motor

• Corresponding Author.

E-mail: neomicro@dreamwiz.com

TEL: +82-51-620-1606; FAX: +82-51-621-1411

Department of Mechanical Engineering, College,

Pukyong National University Korea (Manuscript

Re-ceived May 15,2001; Revised October 26, 2001)

J

KDp

KDs

K1s

Kp p

tc.

Is

r.

Po

rotor on driving wheel axis : Inertia moment of rotor : Derivative gain for the mobile robot : Derivative gain for the torch slider : Integral gain for the torch slider : Proportional gain for the mobile robot : Proportional gain for the torch slider : Maximum distance of the seam tracking sensor

: Maximum distance of the torch slider : Mass of mobile robot excluding masses for driving wheels and rotors of DC motors

: Mass of driving wheel including rotor of motor

: Mass center of the mobile robot with coordinates (xc, Yc)

: Geometric center with coordinates (xo,

Yo), that is the intersection between

symmetry and the driving wheel axis

84 Yang Bae Jean, Sang Bong Kim and Soon Sit Park

r» :Radius of pinion

Y : Radius of driving wheel

Vweld : Welding speed

xs :Distance of the seam tracking sensor

Xts : Distance of the torch slider

Xtss : Distance of the end of torch

X - Y :World coordinate system

x-y :Coordinate system fixed on the mobile

robot

Greeks:

8sm : Motor shaft angle

rp :Torque acting on the left and right

wheel

rs :Torque acting on the torch slider

1 Introduction

Usually, in welding process of the shipbuilding

industry, ship bottom is assembled of several egg

box type of blocks in order to enhance intensity

The egg box is completed by welding processes of

horizontal, vertical and lattice types Since the

welding process is very complicated, it mainly

depends on worker's experience To realize an

automatic welding process, in the case of using a

manipulator type of welding robot, we can not

avoid from several problems such as finding a

slowly start welding point, mobility, cost,

miniaturization, and so on

Nowadays, as a method for automatic welding,

a mobile type of welding robot is employed for

welding line of horizontal type (Kang, C J et al.,

2000), but it can not weld the lattice type of

welding line Usually, the corner part in the

lattice had been welded by worker's hand Since

the working space is very narrow, the welding

workers need robots with lightly weight and small

size Thus, the conventional 6 degrees-of-freedom

(DOF) robots are not appropriate for the lattice

welding Therefore, in order to realize more

compactly automatic welding under complicate

welding environment, an intelligent type of

welding robot with small size and lightly weight

is needed to be developed

Wheeled mobile robots (WMR) constitute a

class of mechanical systems characterized by

kinematic constraints that are not integrable and can not be eliminated from the model equations (dAndrea-Novel et al., 1991, Fierro and Lewis,

1995, Yun and Yamamoto, 1993) Thus, the standard planning and control algorithms developed for usual robotic manipulators without constraints are no more applicable The modeling issue of the WMR for the motion planning and control design is still a relevant question Campion et al analyzed the structural properties and classification of kinematic and dynamical models of the WMR to give a general and unifying presentation of the modeling issue of the WMR (d.Andrea-Novel et al., 1991, Campion et al., 1996) They took into account the restriction

to the robot mobility induced by the constraints, and partitioned 5 classed by introducing the concepts of degree of mobility and manipulation Most of efforts related to the mobile robot control are concentrated on the mobile manipulator that typically consists of a mobile platform and a robotic manipulator mounted upon the platform (Kang, J G et al., 2000, Yamamoto and Yun, 1999) Thus, coordination of manipulator and locomotion is one of the main research topics of the mobile manipulators The majority of the early works on the mobile manipulators focuses

on the coordination of locomotion and manipu-lation by considering the manipulator and the platform as two independent entities (Chung and Hong, 1999, Chung and Velinsky, 1999, Yamamoto and Yun, 1994) Also, they do not take the interactions with the environment into account

In the case of a mobile robot for welding purposes, there are very complex problems such that the motion control must be done in the view

of keeping constant welding speed and precise target line even though the robot is driven for following straight line or corner To obtain good welding bead, the welding speed must be kept constant or at least in a predefined limited range Furthermore, the position of the mobile robot must be controlled to asymptotically converge because of a limited length of torch slider In addition, a slider of the mobile robot carrying torch must be controlled for the end of torch to be

installed for point and it is

Fig 2 Configuration of torch slider

"~ X'==1

n J~aur

/'''''- , x / ,." < , ">,

Fig 1 Motion geometry of a mobile robot

attached at the front side of the body

vii. An electric magnet is set up at the bottom

of robot's center in order to enhance driving force

viii. The mobile platform can only move in the direction normal to the axis of the driving wheels

ix. The velocity component at the point contacted with the ground in the plane of the wheel is zero

x Although tremendous friction force acts on the mobile platform, the two motors have enough power to move it

xi. The mobile platform is moving on a horizontal plane

xii When the mobile platform is driven at the corner in the lattice space, it turns around one point

where ¢> is the heading angle of the mobile plat-form, and BT, Bl are the angles of the right and left driving wheels, respectively From assumptions

The configuration of the torch slider can be described as shown in the Fig 2

If we ignore the passive wheels, the configura-tion of the mobile platform can be described by five generalized coordinates

2.1 Kinematical constraint equations

In this section, we derive the motion and

con-straint equations of the mobile platform with a

geometrical motion as shown in Fig 1.To get the

kinematical equations and to control the mobile

robot by the proposed methods which will be

stated in the following sections with the following

assumptions

i Robot has two rotating wheels for body

motion control

ii Two driving wheels are positioned on an

axis passed through the vehicle geometric

center

iii Two passive wheels (castors) are installed

at the bottom of front and rear for balance of

mobile platform

iv A torch slider is located at the center of

mobile robot and is composed of rack and

pinion gear

v A seam tracking sensor is located at the

upper side of torch and a compensating

sensor is attached at the rear side of body,

where two sensors are made of linear

potentiometers

vi A proximity sensor is

detection of corner rotation

2 Modeling for Mobile Robot

kept at the welding target line

In this paper, the mobile robot is modeled

based on Lagrange equation under nonholonomic

constraints and the model is represented in the

state space form To solve the above problems,

three types of control algorithms for the welding

mobile robot are suggested: straight locomotion,

seam tracking and turning locomotion controls A

concept of decoupling method between input and

output is adopted for the straight locomotion The

PID control method is used for the torch slider

control to seam tracking, and for the turning

locomotion The turning speed is controlled by

the angular velocity value at each point of the

corner with range of 90° constrained to the

welding speed Simulations have been done to

verify the effectiveness of the proposed control

systems

86 Yang Bae Jean, Sang Bong Kim and Soon Sil Park

Vlll and ix, we can get the three constraints as

follows First, the velocity of the pointPsmust be

directed in the direction of the symmetry axis The

relation of velocity around Pccan be expressed as

follows:

=AlCOS ¢+(Az+ Ila)sin¢

IwrJr=fr-Azrw IwrJl=rz-llarw

(9) ( 10) (II) ( 12)

Rearranging the above stated three constraints

can be written in the form of

The other two constraints are obtained by the

equations related to the velocities as follows :

Xc cos¢+ycsin¢+b¢=rwBr (3)

where AI, Az, Ila are Lagrange multipliers corre-sponding to 3 independent kinematical

constraints t-, t, are the torques acting on the right and left wheels, respectively These five equations describing the motion of the mobile robot can easily be written by the following vector form

where (5)

A(q)q=O

where

m=mc+2mw

)+2Im

r

M(q)= mwdsin¢ -mwdcos¢ I 0 0

V(q, q)=r~:;::~l EI'lJ

0 0 1

2.3 State space representation

To transform the above dynamic equation into the state space form, let us define thatS(q) is the null space of A(q) so as to remove Lagrange multipliers S(q) is given by

r

db cos r/J-d sinr/J) db cos r/J+d sinr/J)1

db sin r/J+d cosr/J) c'bsin r/J-d cosr/J)

r w C=TJi'

As the constraint Eq (5) is zero, we can see thatqis in the null space ofA(q). It follows that

qEsPan{sl(q), sz(q)}, and it is possible to express qas a linear combination ofSI(q) and S2

(q), i.e.,

(8)

=AISin ¢+(Az+ Ila)cos¢

where

2.2 Dynamic equations of motion

The potential energy is zero (V=O) since It IS

assumed that the mobile platform is moving on a

horizontal plane The friction energy can be

regarded as zero (F=O) from assumptions Thus,

the total kinetic energy T of the mobile robot is

given by

[

- s in ¢ cos¢ - d 0 0 -,

rw-It is easy to check that' A(q) has rank 3

Consequently, the mobile platform has two DOF

To derive the dynamic equation for the mobile

robot, we apply the well known Lagrange

equa-tion for nonholonomic constraints to the moequa-tion

of the mobile platform as follows:

For the specific choice of the matrix S(q) in

Eq (14), we have 7]=fJ, where fJ=[fJ r fJlF.

Now, let us multiply ST(q) to both sides of the

dynamic Eq (13), then, we have

and

-cJt= x tssCOS 'f' - Xtss'f'Sill1>

where Vc is the forward velocity of the mobile

robot In Fig 2, by appling the Newton's Second Law to the rotor, we can get the following equa-tion

(22)

(2l) Now, let us multiply radius of pinion at both sides of above equation and substitute its for Yp

~ and Xts lor Yp~because r» sm is the length of torch slider(Xts) Then, we have

(17)

( 18)

ST(q)M(q) (S(q) i;+S(q) 7])

+ST (q) V(q, q)= rp

=ST(q)£(q) rp-ST(q)AT(q)A

Using ST(q)AT(q)=0 and ST(q)£(q)=

lzxz, and substituting the Eq (16) for the above

equation, we can obtain

Using the state space variables,x= [xc Yc1> ar

al fJr fJl] T, the dynamics of the mobile platform

can be represented in the state space form:

where

The distance of the seam tracking sensor, Xs

shown in Fig 2, can be calculated by

x= [Xl Xz X3 X4 Xs X6 X7 XsXg XIOXu] T

= [Xc Yc 1> ar al fJr fJl x., Xtsx, Vweld] T,

r=[rrt, ts]T.

Then, the DOF of the mobile robot is three

Xs={ s~a1>-Xts=I (Xa, xe. 1» : Os'xssls, (23)

Is : xs> Is

The seam tracking sensor has a spring for making initial distance of the seam tracking sensor Thus, if the value x, is less than the maximum length, then, Xs can be calculated by

Eq (23) Whilex, is lager than maximum length,

x, is set by the maximum length (/,J.

Now, by including the four state variables xu,

Xts xs, Vweld into Eq (19), we can obtain the

augmented state equation with all states for the mobile platform and torch slider as follows:

To control the welding speed, first we must get

the welding speed In Fig 3, when the mobile

robot moves from (i- I)th position to (i) th

posi-tion, the welding speed is calculated as follows :

=Xtsscos 1>-Xtss¢sin 1>+Vcsin 1>=v (q)

where

PaPe=Xtsssin (90-1»,

y

x

Fig 3 Motion of the mobile platform

-x=l-(S'MS}~;:$'+S'VI +

i(xo: Xs X3)

v(q)

where

o

-(STMS)-l

o

o o

o

o _

o

o

r

em

o

o -(24)

88 Yang Bae Jeon, Sang Bong Kim and Soon Sil Park

and the forward velocity of the mobile platform is given by

(33)

The decoupling matrix for this output equation

is computed as follows (Sarkar et al., 1994, Shankar, 1999) :

YPI aUhPl~~S(q)]7J+JhPl(q)S(q)up (36)

of the robot shown in the output equation:

where hPl (q) is defined as the shortest distance from pointPc of mass center to the desired path,

platform To consider a straight line path, let the path be described by Px+Qy+R=O. Thus, we can derive the shortest distance, hpl (q) for the above path

(27)

where

Let us define the control input as follows:

where Up is the control input for the mobile

platform and Usis the control input for the slider

Then, the state equation can be simplified to the

form:

because of added freedom of the torch slider For

the number of actuator inputs is equal to the DOF

of the mobile robot, we can apply the following

nonlinear feedback control for the mobile

plat-form:

where

where

(40) (39) (38)

I

t.:

Because (j)is bounded away from zero for allx,

we can derive the control input for the straight locomotion in Eq (28) as follows:

The output equation for forward velocity of the mobile platform can be given by

YP2= aq-x=JhP2 q) Up

Therefore, the decoupling matrix is yielded as

where

(30)

3.2 Straight locomotion control

To control the welding speed, we control the

velocity of the mobile platform As the mobile

platform has two motors, we may choose two

output variables to control position and velocity

3 Control Algorithms

3.1 Torch slider control

To control the torch slider for seam tracking, a

PID controller method is used We may choose

the following output equation :

The tracking error for the seam tracking sensor

is defined as follows:

The control input for the torch slider in Eq

(28) is designed by using the PID controller:

Then, the path errors and forward velocity of

the mobile robot are defined as follows: [ epJ = [V~-YPIJ.

Table 1 Numerical values of the mobile robot

Parameters Values Units Parameters Values Units

3.3 Turning locomotion control

A proximity sensor detects the rotation point at the corner, then, the robot rotates the corner for welding and its sliding arm is controlled for the end of torch to be kept at the welding target line When the robot is driven at the corner in the lattice space, the left and right wheels are driven

in the opposite direction The absolute speed of two wheels is exactly equal In addition, the electric magnet prevents to stray away from

turn-257.5r r .,. -r -r .,. -r ~

68

60

50

- - - - simulation

- - - reference

30 40

Y position (mm)

20

257.0

o 10

(b) The position Xc

~255.5

10

- - - reference

- - - - simulation

TIme(s)

(a) The welding speedVweld

50

-;;-Ii 0 • _ •• • "-~ -J

!

] -50

~"00

:!;!.150 V

~

-200

-250 ' - - - - ' - - - - ' - - - ' - - ' - _ ' - - -' -.1

o

186

10

10 7

- - - - simulation

- - - reference

4 5

Time (s)

5 6

Time (s)

(d) Distance of the seam tracking sensor x,

5r -,. , .,.-"""' ' "' "' " ' -'

4

",'

'" 0 - - - 1

~ -2

~ ~

·5

.;ll - _-'-_.J. -'_-'-_-'-_' _-' '

66

~ 85

!. 84

\

~ 82

:Q

<l

g 81

~

"l so

79

9 10

- - - - left motor

- - - rlghtmotor

5

Time (s)

2

2

,',

f \

\ > c : = = - - - l

TIme(s)

(c) Control input for mobile robot Up

-20!

.25 L -'_~ _ _' _ " _ _ J _ _ ' _ _ _ _' -'- .J

o

i'184

-!.182

~180

~ 178

.0>

~ 178

174N

172 '-~ ' -'-_" -'_ _-'-_-'- .J

o

25 , - , - - , - - , - - , - , - - , - - , - - - , _ - , - _ ,

20

15

:I' 10

'" 5

.~ 0

] ·5

a-10

·15 {

(e) Distance of the torch sliderx« (f) Control input for torch slider Us

Fig 5 Simulation results of straight locomotion

90 Yang Bae Jeon, Sang Bong Kim and Soon Sit Park

Fig 4 Block diagram of the closed loop system

The error for angular velocity is defined by :

line and curved line In simulation, it is assumed that disturbance and noise do not affect the sys-tem The numerical values of the system parameters used in the simulations are given in Table I

We considered a straight line path, x =255

mm, as shown in Fig 5 (a) to give reality of the welding at the lattice space The initial position of the robot is (xc, Yc)=(257mm, Omm) , the heading angle is 1>=80°. And, we assumed that the length of torch slider is initialized always

Xts= 175mm. Then, the initial distance of the

seam trac mg sensor ecomes Xs= cos (100)

-Xts } =85.964mm. Usually, to obtain a good welding bead, the welding speed is chosen as

about 7.5mm/sin the case of using an arc welder Thus, we take the above stated speed for the reference welding speed In part of turning locomotion control, we already assumed that the mobile robot is turning around one point Thus, the forward velocity of the mobile platform is set

to be zero As the reference welding speed is

Vweld= 7.5 mm/Sand the turning position is x= 255mm, we can calculate the reference angular velocity of the mobile robot as shown in Fig 6

(b) The initial length of the torch slider is Xts= 175mm. And, the PID gains were determined by repeated simulation results The initial values of the mobile robot for simulations are shown in Table 2

The simulation results for straight locomotion and turning locomotion are shown in Figs 5-6 The operation of the mobile robot can be stated

as follows: first, the mobile robot will track the start welding position and next, the welding pro-cess begins In Fig 5 (a), the mobile robot tracks its start welding position in 5 seconds There is no welding process when the robot is tracking its start welding position Thus, the welding speed is

no meaning at this time that is setting the initial welding process After about 5 seconds, the mobile robot starts to weld, tracks well the welding line and the welding speed is kept con-stantly for the reference velocity Also, the control

of the seam tracking sensor is well done as shown in

(43)

(47)

(46)

(45)

sin ¢2Vwetd

Xo

ing point Thus, we already assumed that the

forward velocity of the mobile platform is zero

By using Eq (20) and the assumption, we can

derive the welding speed as follows:

Vwetd=PaPe' =Xtsscos 1>-Xtss¢sin¢

When the robot turns, the initial point of the

robot may be invariable in time(Xo= constant) ,

from the assumption Then, we can derive a

simple equation and the relation between welding

speed and angular velocity of the robot:

-J,2

Xo

We consider a trajectory consisted of a straight

4 Simulation Results

Figure 4 describes the feedback loop control

algorithm incorporating the 3 cases of the robot

control The straight locomotion and the turning

locomotion are controlled case by case, but torch

slider control works well always In the figure,x"

is the reference value for each controller, and eis

the error value for each output

Using the above equation, when the mobile

robot is turning at the lattice space, the control

input for two wheels of the mobile robot can be

given by :

Then, we may choose the following output

equation:

Table 2 Condition values for simulations

· Initial (xc, Yc) (xc, Yc) = (257mm, Omm) (xc, Yc) = (255mm, Omm)

· Initial Uc vc=Omm/s vc=Omm/s

· Initial Xts xts=175mm xts=175mm

· Initial Xs xs=85.964mm xs=80mm

· Output equation hs(x) =Xs, h p1 (q) = (xc-255) , hs(x) =Xs,

h p2(7])= ~w (7]1+7]2) hp(x)=¢

· Reference input xt=80, vt=O, vt=7.5 xt=80, ¢d= 7.5sin¢2

255

Gain for the robot K pp=2.5, K Ds=2.95 Kpp=IO, KDs=IOO

(Feedback gains)

Gain for the torch slider K ps= 1700, K1s=0.1, K Ds=690 K Ps=1700, K1s=0.1, K Ds=690 (PlD gains)

Sampling time LlT=O.Ols LlT=O.Ols

- - - - simulation

- - - reference

.30 = _ J - _ - - - '_ _- ' - _ - - - '_ _ -'-_ ' :=

80 68 50

- - - - simulation

- - - reference

20

Time (s)

(b) The angular velocity ¢ (a) The welding speed Uweta

10.0 r-~ ' """-" ""' -'-"""-""""-",,, ,

9.0

~ ;~· · - · · · 7 · - - - 1

~

~ 5.0

i 4.0

~ 3.0

"" 2.0

1.0

0.0 - - ' _ _ - ' - _ _ ' - - - - ' - ' - _ J - ~

o 0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0

Time (s)

- - - - simulation

- - - reference

3~\

, 2 ,

~

1;

~

v

]

~

a ·2

·3

- - - - I e j l motor

- - - right motor

80.010, , ., , , ., - _,. -,

eo.OO8

i'80

-! 006

~80.0004

~eoOO2V_ -_ _ -'l

",eo.OOO

' . -. - -.-.- - -.-.-~79.998

g 79.996

~79._

~19.992

60 68 50

20

'0 30 40

Time(s)

(d) Distance of the seam tracking sensorx,

79.990 - - - - ' - - - - ' - - - - -'-~

o

4.0 r -r ., -~ , , _- _,.- ,

a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

(C) Control input for mobile robot Up

300 , - - , - - - - , - - - , - - - - , - - - , - - - , - - ,

3.5

.3.0

~

1; 2.5

.~ 2.0

'2 1.5

"

~ 1.0

0.5 0

Time (s)

50 68 50

20

Time (s) (e) Distance of the torch slider Xts (f) Control input for torch slider Us

Fig 6 Simulation results of turning locomotion

180 L - _ _ _-' _-'-_ _-'-_-'-_ _-' '

o

180

92 Yang Bae Jean, Sang Bong Kim and Soon Sil Park

Fig 5 In simulation results for turning

locomotion, there is a little error for the seam

tracking sensor as shown in Fig 6 (d) , but it is no

affected for welding at the corner because the

maximum error is about0.002mm. InFig 6 (a),

the mobile robot tracks well the reference angular

velocity and the welding speed is kept constantly

for the reference velocity

5 Conclusion

This paper introduced a motion control method

of the mobile robot for the lattice type of welding

line, and proved the possibility that the mobile

robot can weld the lattice type welding line We

have proposed the separated control algorithms

for straight locomotion, seam tracking and

turn-ing motion The straight locomotion control

sys-tem design is done by using the dynamic

nonlinear state feedback and the nonlinear state

transformation which decouple the dynamic

equations of the mobile platform The PID

con-troller method is employed for seam tracking In

addition, we have designed a turning motion

controller by using the relating equation between

angular velocity of the robot and given welding

speed Simulations have been done in two cases:

the mobile robot welds along straight line and

curved line Through the simulation results, it can

be said that the welding speed depends on initial

position and initial heading angle of the mobile

robot Moreover, each gain value affects tracking

time of position and welding speed The results

have proved that this system has enough ability to

weld the lattice type welding line when the mobile

robot is equipped for the division of the

shipbuilding industry that needs the lattice type

welding line It is alone expected that these results

can be effectively used to control a real system for

future works

Acknowledgement

This paper is a part of a study titled

"Develop-ment of Mobile Robot for Lattice Type Welding

by Using Arc-sensor" which is studied by

Minis-try of Commerce, IndusMinis-try, and Energy support

We gratefully acknowledge the contributions and suggestions of related persons

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